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Golay Pair
: ''For complementary sequences in biology, see complementarity (molecular biology). For integer sequences with complementary sets of members see Lambek–Moser theorem.'' In applied mathematics, complementary sequences (CS) are pairs of sequences with the useful property that their out-of-phase aperiodic autocorrelation coefficients sum to zero. Binary complementary sequences were first introduced by Marcel J. E. Golay in 1949. In 1961–1962 Golay gave several methods for constructing sequences of length 2''N'' and gave examples of complementary sequences of lengths 10 and 26. In 1974 R. J. Turyn gave a method for constructing sequences of length ''mn'' from sequences of lengths ''m'' and ''n'' which allows the construction of sequences of any length of the form 2''N''10''K''26''M''. Later the theory of complementary sequences was generalized by other authors to polyphase complementary sequences, multilevel complementary sequences, and arbitrary complex complementary sequences. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complementarity (molecular Biology)
In molecular biology, complementarity describes a relationship between two structures each following the lock-and-key principle. In nature complementarity is the base principle of DNA replication and transcription as it is a property shared between two DNA or RNA sequences, such that when they are aligned antiparallel to each other, the nucleotide bases at each position in the sequences will be complementary, much like looking in the mirror and seeing the reverse of things. This complementary base pairing allows cells to copy information from one generation to another and even find and repair damage to the information stored in the sequences. The degree of complementarity between two nucleic acid strands may vary, from complete complementarity (each nucleotide is across from its opposite) to no complementarity (each nucleotide is not across from its opposite) and determines the stability of the sequences to be together. Furthermore, various DNA repair functions as well as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wi-Fi
Wi-Fi () is a family of wireless network protocols, based on the IEEE 802.11 family of standards, which are commonly used for local area networking of devices and Internet access, allowing nearby digital devices to exchange data by radio waves. These are the most widely used computer networks in the world, used globally in home and small office networks to link desktop and laptop computers, tablet computers, smartphones, smart TVs, printers, and smart speakers together and to a wireless router to connect them to the Internet, and in wireless access points in public places like coffee shops, hotels, libraries and airports to provide visitors with Internet access for their mobile devices. ''Wi-Fi'' is a trademark of the non-profit Wi-Fi Alliance, which restricts the use of the term ''Wi-Fi Certified'' to products that successfully complete interoperability certification testing. the Wi-Fi Alliance consisted of more than 800 companies from around the world. over 3.05 billion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zadoff–Chu Sequence
A Zadoff–Chu (ZC) sequence, also referred to as Chu sequence or Frank–Zadoff–Chu (FZC) sequence, is a complex-valued mathematical sequence which, when applied to a signal, gives rise to a new signal of constant amplitude. When cyclically shifted versions of a Zadoff–Chu sequence are imposed upon a signal the resulting set of signals detected at the receiver are uncorrelated with one another. They are named after Solomon A. Zadoff, David C. Chu and Robert L. Frank. Description Zadoff–Chu sequences exhibit the useful property that cyclically shifted versions of themselves are orthogonal to one another. A generated Zadoff–Chu sequence that has not been shifted is known as a ''root sequence''. The complex value at each position ''n'' of each root Zadoff–Chu sequence parametrised by ''u'' is given by : x_u(n)=\text\left(-j\frac\right), \, where : 0 \le n < N_\text, : and , : |
Hadamard Code
The Hadamard code is an error-correcting code named after Jacques Hadamard that is used for error detection and correction when transmitting messages over very noisy or unreliable channels. In 1971, the code was used to transmit photos of Mars back to Earth from the NASA space probe Mariner 9. Because of its unique mathematical properties, the Hadamard code is not only used by engineers, but also intensely studied in coding theory, mathematics, and theoretical computer science. The Hadamard code is also known under the names Walsh code, Walsh family, and Walsh–Hadamard code in recognition of the American mathematician Joseph Leonard Walsh. The Hadamard code is an example of a linear code of length 2^m over a binary alphabet. Unfortunately, this term is somewhat ambiguous as some references assume a message length k = m while others assume a message length of k = m+1. In this article, the first case is called the Hadamard code while the second is called the augmented Hadamar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Golay Code
In coding theory, the ternary Golay codes are two closely related error-correcting codes. The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distance of the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is a perfect code. The extended ternary Golay code is a 2, 6, 6linear code obtained by adding a zero-sum check digit to the 1, 6, 5code. In finite group theory, the extended ternary Golay code is sometimes referred to as the ternary Golay code. Properties Ternary Golay code The ternary Golay code consists of 36 = 729 codewords. Its parity check matrix is : \left[ \begin 1 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 0\\ 1 & 2 & 1 & 0 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\ 1 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \end \right]. Any two different co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Length Sequence
A maximum length sequence (MLS) is a type of pseudorandom binary sequence. They are bit sequences generated using maximal linear-feedback shift registers and are so called because they are periodic and reproduce every binary sequence (except the zero vector) that can be represented by the shift registers (i.e., for length-''m'' registers they produce a sequence of length 2''m'' − 1). An MLS is also sometimes called an n-sequence or an m-sequence. MLSs are spectrally flat, with the exception of a near-zero DC term. These sequences may be represented as coefficients of irreducible polynomials in a polynomial ring over Z/2Z. Practical applications for MLS include measuring impulse responses (e.g., of room reverberation or arrival times from towed sources in the ocean). They are also used as a basis for deriving pseudo-random sequences in digital communication systems that employ direct-sequence spread spectrum and frequency-hopping spread spectrum transmission sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Binary Sequence
A pseudorandom binary sequence (PRBS), pseudorandom binary code or pseudorandom bitstream is a binary sequence that, while generated with a deterministic algorithm, is difficult to predict and exhibits statistical behavior similar to a truly random sequence. PRBS generators are used in telecommunication, such as in analog-to-information conversion, but also in encryption, simulation, correlation technique and time-of-flight spectroscopy. The most common example is the maximum length sequence generated by a (maximal) linear feedback shift register (LFSR). Other examples are Gold sequences (used in CDMA and GPS), Kasami sequences and JPL sequences, all based on LFSRs. In telecommunications, pseudorandom binary sequences are known as pseudorandom noise codes (PN or PRN codes) due to their application as pseudorandom noise. Details A binary sequence (BS) is a sequence a_0,\ldots, a_ of N bits, i.e. :a_j\in \ for j=0,1,...,N-1. A BS consists of m=\sum a_j ones and N-m zeros. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyphase Sequence
In mathematics, a polyphase sequence is a sequence whose terms are complex roots of unity: : a_n = e^ \, where ''x''''n'' is an integer. Polyphase sequences are an important class of sequences and play important roles in synchronizing sequence design. See also *Zadoff–Chu sequence A Zadoff–Chu (ZC) sequence, also referred to as Chu sequence or Frank–Zadoff–Chu (FZC) sequence, is a complex-valued mathematical sequence which, when applied to a signal, gives rise to a new signal of constant amplitude. When cyclically shi ... References *{{cite book , first=Pingzhi , last=Fan , first2=Michael , last2=Darnell , title=Sequence Design for Communications Applications , publisher=Research Studies Press , year=1996 , isbn=047196557X Sequences and series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kasami Code , a line code associated with him
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Kasami may refer to: *Kasami, Iran, a village in Sistan and Baluchestan Province, Iran *Pajtim Kasami, a Swiss footballer *Tadao Kasami, a Japanese information theorist :*Kasami code Kasami may refer to: * Kasami, Iran, a village in Sistan and Baluchestan Province, Iran * Pajtim Kasami, a Swiss footballer * Tadao Kasami, a Japanese information theorist :* Kasami code, a line code associated with him {{dab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gold Code
A Gold code, also known as Gold sequence, is a type of binary sequence, used in telecommunication (CDMA) and satellite navigation (GPS). Gold codes are named after Robert Gold. Gold codes have bounded small cross-correlations within a set, which is useful when multiple devices are broadcasting in the same frequency range. A set of Gold code sequences consists of 2''n'' + 1 sequences each one with a period of 2''n'' − 1. A set of Gold codes can be generated with the following steps. Pick two maximum length sequences of the same length 2''n'' − 1 such that their absolute cross-correlation is less than or equal to 2(''n''+2)/2, where ''n'' is the size of the linear-feedback shift register used to generate the maximum length sequence (Gold '67). The set of the 2''n'' − 1 exclusive-ors of the two sequences in their various phases (i.e. translated into all relative positions) together with the two maximum length sequences form a set of 2''n'' + 1 Gold code sequenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error-correcting Code
In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is the sender encodes the message with redundant information in the form of an ECC. The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. ECC contrasts with error detection in that errors that are encountered can be corrected, not simply detected. The advantage is that a system using ECC does not require a reverse channel to request retransmission of data when an error occurs. The downside is that there is a fixed overhead that is added to the message, thereby requiring a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Golay Code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory. There are two closely related binary Golay codes. The extended binary Golay code, ''G''24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, ''G''23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
